Here \(\mathbb {C}_{13}\)  is the
metric contraction operator in the
1st and 3rd entries. See p. 83 in
[1]}.
The divergence of \(\mathcal {S}(f)\)  is
given explicitly by [2]}:
\(\mathrm {div} \: \mathcal {S}(f)=-g_{P}(\tau (f),d f).\) 
